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GS Formalism
GS String Theory, or the GS Superstring Theory was an early attempt to include fermions in String Theory. "GS" stands for Green-Schwarz. In contrast to the RNS String Theory, the GS String Theory automatically has spacetime supersymmetry, but not worldsheet supersymmetry. Introduction to Spacetime Supersymmetry Instead of the “fake” fermionic fields in the RNS String Theory, which are really spacetime vectors, the GS String Theory has “actual” bosonic and fermionic fields {X^\mu },\Theta . The Spacetime Supersymmetric transformations are given by: \begin{align} & \delta \leftrightarrow \\ & \delta \leftrightarrow }^{A}} \\ \end{align} This is actually intuitively related to the worldsheet supersymmetry of the RNS String Theory. Thsee are also also the transformations of superspace. It is to be noted that here, \gamma^\mu is the Dirac Gamma Matrix on the background spacetime. It is relatively trivial to show that: \begin{align} & }_{1}}, =-2\bar{\varepsilon }_{1}^{A} \varepsilon _{2}^{A}= \\ & }_{1}}, =0 \\ \end{align} This means that the commutator bracket of infinitesimal Supersymmetric transformations, translates the bosonic field by a^\mu , and leaves the fermionic field untouched. These transformations, combined with the stanadard Poincaire transformations, give rise to the Super-Poincaire transformations, forming the Super-Poincaire Group. Clearly, to be consistent with the transformations of superspace, the standard D0 brane action would have to be changed in such a way that is replaced with the field: \Pi _{0}^{\mu }= - ^{A}} The subscript of 0 on the Left-Hand-Side indicates that only time-derivatives are taken. This will be especially clear when we discuss spacetime Supersymmetry for strings and other D-branes. The D0-brane action then becomes: =-m\int_ ^ {\sqrt{- \cdot }\text{d}\tau } This is invariant under super-poincaire transformations (and diffeomorphisms, of course). These D0-branes are the same D0-branes that appear in the Type IIA String Theory. The \mathsf{\mathcal{N}}=2 supersymmetry of the Type IIA String Theory is interpreted as having 2 spinor fermionic coordinates , . Since Type IIA String Theory is a chiral theory, these have opposite chirality. In other words, due to the GSO Projection, we have: \begin{align} & =\frac{1}{2}\left( 1+ \right)\Theta \\ & =\frac{1}{2}\left( 1- \right)\Theta \\ \end{align} Kappa Symmetry and D0 Branes The total (Kappa-Symmetric) action is: ^ {\sqrt{- \cdot }\text{d}\tau }+\int } \Theta \text{d}\tau \right) |cellpadding = 6 |border = 1 |border colour = black |background colour=white}} Spacetime Supersymmetry for Strings The Nambu-Goto Action is given by: S=-T\int_ ^ {\sqrt{-\det \left( \right)} ^{2}}\sigma } In analogy with the action for the Supersymmetric D0 brane, the action for the Supersymmetric string would become: =-T\int_ ^ {\sqrt{-\det \left( \Pi _{\beta }^{\mu } \right)} ^{2}}\sigma } Where we define: \Pi _{\alpha }^{\mu }= - ^{A}} In a way analogous to the Pi mu nought for the D0 - branes. Kappa Symmetry and D1 branes The total action (with \mathcal N=2 supersymmetry) is therefore: ^ {\sqrt{-\det \left( \Pi _{\beta }^{\mu } \right)} ^{2}}\sigma } \\ & \text{ }+\frac{1}{\pi }\int ^{2}}}\sigma \left( }^{1}} }^{1}}\text{ } }^{2}} -i \left( }^{1}} - \right) \right)\text{ } \\ \end{align} |cellpadding = 6 |border = 1 |border colour = black |background colour=white}} (Please click here to view the following properly) Relation with RNS String Theory GS String Theory is equivalent to a GSO Projected RNS String Theory. The specific consistent String Theory that this is equivalent to depends on the chirality and number of components of the fermionic fields. Category:String Theory